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On the dynamic scaling behaviour of solutions to the discrete smoluchowski equations

Published online by Cambridge University Press:  20 January 2009

F. P. Da Costa
F. P. Da Costa
Affiliation:
Instituto Superior Técnico Departamento de Matemática, Av. Rovisco Pais, P-1096 Lisboa Portugal E-mail address: [email protected]
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Abstract

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In this paper we generalize recent results of Kreer and Penrose by showing that solutions to the discrete Smoluchowski equations

with general exponentially decreasing initial data, with density p, have the following asymptotic behaviour

where J = {j: cj(t)>0, t>0} and q = gcd{j: cj(0)>0}.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 39 , Issue 3 , October 1996 , pp. 547 - 559
Copyright
Copyright © Edinburgh Mathematical Society 1996

References

REFERENCES

1. Ball, J. M. and Carr, J., The discrete coagulation-fragmentation equations: existence, uniqueness, and density conservation, J. Statist. Phys. 61 (1990), 203–234.CrossRefGoogle Scholar
2. Ball, J. M., Carr, J. and Penrose, O., The Becker-Döring cluster equations: basic properties and asymptotic behaviour of solutions, Comm. Math. Phys. 104 (1986), 657–692.CrossRefGoogle Scholar
3. Ball, J. M., Holmes, P. J., James, R. D., Pego, R. L. and Swart, P. J., On the dynamics of fine structure, J. Nonlinear Sci. 1 (1991), 17–70.CrossRefGoogle Scholar
4. Carr, J. and Da Costa, F. P., Instantaneous gelation in coagulation dynamics, Z. Angew. Math. Phys. 43 (1992), 974–983.CrossRefGoogle Scholar
5. Carr, J. and Da Costa, F. P., Asymptotic behavior of solutions to the coagulation-fragmentation equations. II. Weak fragmentation, J. Statist. Phys. 77 (1994), 89–123.CrossRefGoogle Scholar
6. Da Costa, F. P., On the positivity of solutions to the Smoluchowski equations, Mathematika 42 (1995), 406–412.CrossRefGoogle Scholar
7. Drake, R., A general mathematical survey of the coagulation equation, in Hidy, G. M. and Brock, J. R. (eds.), Topics in Current Aerosol Research (part 2) (International Reviews in Aerosol Physics and Chemistry, Pergamon Press, Oxford, 1972).Google Scholar
8. Ernst, M. H. and Van Dongen, P. G., Scaling solutions of Smoluchowski's coagulation equations, J. Statist. Phys. 50 (1988), 295–329.Google Scholar
9. Family, F. and Vicsek, T., Dynamic scaling for aggregation of clusters, Phys. Rev. Lett. 52 (1984), 1669–1672.Google Scholar
10. Friedlander, S. and Wang, C., The self-preserving particle size distribution for coagulation by brownian motion, J. Coll. Interface Sci. 22 (1966), 126–132.CrossRefGoogle Scholar
11. Hua, L. K., Introduction to Number Theory (Springer-Verlag, Berlin, 1982).Google Scholar
12. Kreer, M. and Penrose, O., Proof of dynamic scaling in Smoluchowski's coagulation equation with constant kernels, J. Statist. Phys. 74 (1994), 389–407.CrossRefGoogle Scholar
13. Shirvani, M. and Van Roessel, H., The mass-conserving solutions of Smoluchowski's coagulation equation: the general bilinear kernel, Z. Angew. Math. Phys. 43 (1992), 526–535.CrossRefGoogle Scholar
14. Slemrod, M., Trend to equilibrium in the Becker-Döring cluster equations, Nonlinearity 2 (1989), 429–443.CrossRefGoogle Scholar
15. White, W., A global existence theorem for Smoluchowski's coagulation equation, Proc. Amer. Math. Soc. 80 (1980), 273–276.Google Scholar